Set Notation And Venn Diagrams

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Notes

Set Notation Basics

A **set** is a collection of elements, written inside curly braces `{}`.
`ℰ` denotes the **universal set** (all elements under consideration).
`n(A)` is the **number of elements** in set A.
`∈` means 'is an element of'; `∉` means 'is not an element of'.
`∅` is the **empty set** (no elements).
`A ⊆ B` means 'A is a **subset** of B' (all elements of A are in B).
`A ⊈ B` means 'A is **not a subset** of B'.

Set Operations

`A ∪ B` is the **union** of A and B (elements in A or B or both).
`A ∩ B` is the **intersection** of A and B (elements in both A and B).
`A'` is the **complement** of A (elements in ℰ not in A).
`A ∪ B` includes all elements from both sets, counting overlaps only once.
`A ∩ B` contains only the shared elements.
`A'` depends on ℰ; e.g., if ℰ $= {1,2,3,4,5}$ and $A = {1,3}$ , then $A' = {2,4,5}$ .

Venn Diagrams – Structure

A Venn diagram uses a **rectangle** for ℰ and **circles** for sets.
Overlapping circles show **intersections**; non‑overlapping parts show elements unique to each set.
The region inside a circle represents that set; outside the circle is its complement.
Numbers or elements are placed in each region to show counts or members.

Interpreting Venn Diagrams

`A ∪ B` is the **total shaded area** inside either circle (including overlap).
`A ∩ B` is the **overlap region** of the two circles.
`A'` is everything **outside** the A circle (but inside ℰ).
`(A ∪ B)'` is the region outside both circles (complement of union).
`A ∩ B'` is the part of A that is **not** in B.

Problem‑Solving with Venn Diagrams

Use given `n(A)`, `n(B)`, `n(A ∩ B)`, `n(ℰ)` to find missing region counts.
For three sets, work from the **centre** (triple intersection) outward.
Remember: total in ℰ $= sum$ of all regions (including outside all sets).
Form equations using `n(A ∪ $B) = n(A) + n(B)$ – n(A ∩ B)`.
Check that no region has a negative count; if so, adjust assumptions.

Set Notation with Descriptions

Sets can be defined by a rule: `{x : condition}` or `{x | condition}`.
Example: `{x : x is an integer and $1 \le x \le$ 10}` means {1,2,…,10}.
The colon or vertical bar is read as 'such that'.
If no type is given, `x` can be any real number.

Common Exam Tips

Always list elements **without repetition** in union problems.
`n(A ∪ B)` includes elements in both sets only once.
`A' ∩ $B' = (A$ ∪ B)'` (De Morgan's Law).
When shading, carefully identify the region described by the set expression.
Double‑check that your Venn diagram satisfies all given totals.

Two‑Set Venn Diagram with Regions Labelled

Shading A ∪ B

Shading A ∩ B

Three‑Set Venn Diagram Template

Practice questions

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1.ℰ $= {21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, A = {22, 24, 26, 28, 30}, B = {21, 24, 27, 30}$ . List the members of A ∩ B.
Easy
- A{24, 30}
- B{22, 24, 26, 28, 30}
- C{21, 24, 27, 30}
- D{21, 22, 24, 27, 30}
2.ℰ $= {21, 22, 23, 24, 25, 26, 27, 28, 29, 30}, A = {22, 24, 26, 28, 30}$ . List the members of A'.
Easy
- A{21, 23, 25, 27, 29}
- B{22, 24, 26, 28, 30}
- C{21, 22, 23, 24, 25, 26, 27, 28, 29, 30}
- D{23, 25, 27, 29}
3.ℰ $= {letters$ of the $alphabet}, B = {b, r, a, z, i, l}, I = {i, r, e, l, a, n, d}$ . List the members of B ∪ I.
Easy
- A{b, r, a, z, i, l, e, n, d}
- B{b, r, a, z, i, l}
- C{i, r, e, l, a, n, d}
- D{a, i, l, r}
4.ℰ $= {letters$ of the $alphabet}, B = {b, r, a, z, i, l}, I = {i, r, e, l, a, n, d}$ . List the members of B ∩ I'.
Easy
- A{b, z}
- B{a, i, l, r}
- C{e, n, d}
- D{b, r, a, z, i, l}
5. $B = {b, l, u, e}, G = {g, r, e, y}, W = {w, h, i, t, e}$ . List all the members of B ∪ G.
Easy
- A{b, l, u, e, g, r, y}
- B{b, l, u, e}
- C{g, r, e, y}
- D{e}
6. $B = {b, l, u, e}, G = {g, r, e, y}, W = {w, h, i, t, e}$ . List all the members of W ∩ G'.
Easy
- A{w, h, i, t}
- B{e}
- C{w, h, i, t, e}
- D{g, r, y}
7.Which of the following is the correct set notation for the empty set?
Easy
- A∅
- Bℰ
- C0
- D{}
8.If ℰ $= {1, 2, 3, 4, 5}$ and $A = {1, 3}$ , what is A'?
Easy
- A{2, 4, 5}
- B{1, 3}
- C{1, 2, 3, 4, 5}
- D{2, 4}

Summary

## Set Notation & Venn Diagrams ### Key Definitions - **Set**: collection of elements (numbers, letters, etc.) written in curly brackets `{}`. - **Universal set (ℰ)**: everything under consideration. - **Empty set (∅)**: set with no elements. - **n(A)**: number of elements in set A. ### Set Notation - `∈` : element of (e.g., 3 ∈ {1,2,3}) - `∉` : not an element of - `A ⊆ B` : A is a subset of B (all elements of A are in B) - `A ∩ B` : intersection (elements in both A and B) - `A ∪ B` : union (elements in A or B or both) - `A'` : complement (elements in ℰ not in A) ### Venn Diagrams - Rectangle = ℰ, circles = sets. - Overlap = intersection. - Outside circles = complement. ### Worked Example ℰ = {1,2,3,4,5,6,7,8}, A = {2,6,12,14,28}, B = {7,14,21,28,35} - n(A) = 5 - {14,28} = A ∩ B - A ∪ B = {2,6,7,12,14,21,28,35} - A' ∩ B' = {1,5,8} (not empty) ### Tips - For union, list each element once. - Complement is everything outside the set. - Use set notation to describe regions (e.g., `M ∩ D'`).

Key terms

Set: A collection of elements, written in curly brackets {}.
Universal set (ℰ): The set of all elements under consideration.
Empty set (∅): The set with no elements.
n(A): The number of elements in set A.
∈: Is an element of.
∉: Is not an element of.
A ⊆ B: A is a subset of B; all elements of A are also in B.
A ∩ B: Intersection of A and B; elements in both sets.
A ∪ B: Union of A and B; elements in A or B or both.
A': Complement of A; all elements in ℰ not in A.
Venn diagram: A diagram with a rectangle (ℰ) and circles (sets) showing relationships.

Slides

Set Notation Basics

A set is a collection of elements, e.g. {1, 2, 3}.
ℰ is the universal set (everything we consider).
n(A) is the number of elements in set A.
∈ means 'is an element of'; ∉ means 'is not an element of'.

If ℰ $= {1,2,3,4,5}$ and $A = {1,3,5}$ , what is n(A)?

Union and Intersection

A ∪ B means the union of A and B: elements in A or B (or both).
A ∩ B means the intersection of A and B: elements in both A and B.
Example: $A = {1,2,3}, B = {2,3,4}$ → A ∪ $B = {1,2,3,4}, A$ ∩ $B = {2,3}$ .

Which of these are true?

If $A = {a,b}$ and $B = {b,c}$ , then A ∩ $B = {b}$ .
If $A = {1,2}$ and $B = {3,4}$ , then A ∪ $B = {1,2,3,4}$ .
The union of two sets is always larger than the intersection.

Complement and Empty Set

A' (complement of A) is all elements in ℰ not in A.
∅ is the empty set, containing no elements.
Example: ℰ $= {1,2,3,4,5}, A = {1,3}$ → $A' = {2,4,5}$ .

If ℰ $= {a,b,c,d,e}$ and $A = {a,c,e}$ , what is A'?

{b,d}
{a,c,e}
{b,c,d}
{a,b,d}

Subset Notation

A ⊆ B means A is a subset of B (all elements of A are in B).
A ⊈ B means A is not a subset of B.
Example: {1,2} ⊆ {1,2,3} is true; {4} ⊆ {1,2,3} is false.

If every element of A is also in B, we write $A ____ B$ .

Venn Diagrams

A Venn diagram uses a rectangle for ℰ and circles for sets.
Overlapping regions show intersections.
The region outside all circles represents elements not in any set.

In a Venn diagram, which region represents A ∩ B?

The left circle only
The right circle only
The overlap of the circles
The area outside both circles

Finding n(A) from a Venn Diagram

Add all numbers inside the circle for set A.
Include numbers in the overlap and the part of A not overlapping.
Example: If A has 5 in the 'only A' part and 3 in the overlap, $n(A)=8$ .

In a Venn diagram, the number of elements in A is written $as ____$ .

Shading Regions

Shade A ∪ B by shading all parts of both circles.
Shade A ∩ B by shading only the overlap.
Shade A' by shading everything outside the A circle.

Which shadings are correct?

Shading A ∪ B means shading both circles entirely.
Shading A ∩ B means shading only the overlapping part.
Shading A' means shading only the A circle.

Worked Example: Using Notation

Given ℰ $= {21,...,30}, A = {22,24,26,28,30}, B = {21,24,27,30}$ .
A ∩ $B = {24,30} (elements$ in both).
$A' = {21,23,25,27,29} (elements$ in ℰ not in A).

Using the sets above, what is A ∪ B?

{24,30}
{21,22,24,26,27,28,30}
{21,22,24,26,27,28,29,30}
{22,24,26,28,30}

Problem Solving with Venn Diagrams

Use given numbers to fill in each region step by step.
Start with the intersection (overlap) if known.
Then find 'only' parts by subtracting overlaps.

If n(ℰ)=30, $n(A)=10, n(B)=13$ , and n(A∪B)'=3, then n(A∩B)=____.

Summary

Know the symbols: ∪, ∩, ', ⊆, ∅, n(A), ∈, ∉.
Venn diagrams help visualise sets and their relationships.
Practice with past paper questions to master the topic.

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